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arXiv · 2508.18845

Unique Decoding of Extended Subcodes of GRS Codes Using Error-Correcting Pairs

Yang Li·Zhenliang Lu·San Ling·Shixin Zhu·Kwok Yan Lam
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Abstract

Extended Han-Zhang codes are a class of linear codes where each code is either a non-generalized Reed-Solomon (non-GRS) maximum distance separable (MDS) code or a near MDS (NMDS) code. They have important applications in communication, cryptography, and storage systems. While many algebraic properties and explicit constructions of extended Han-Zhang codes have been well studied in the literature, their decoding has been unexplored. In this paper, we focus on their decoding problems in terms of $\ell$-error-correcting pairs ($\ell$-ECPs) and deep holes. On the one hand, we determine the existence and specific forms of their $\ell$-ECPs, and further present an explicit decoding algorithm for extended Han-Zhang codes based on these $\ell$-ECPs, which can correct up to $\ell$ errors in polynomial time, with $\ell$ about half of the minimum distance. On the other hand, we determine the covering radius of extended Han-Zhang codes and characterize two classes of their deep holes, which are closely related to the maximum-likelihood decoding method. By employing these deep holes, we also construct more non-GRS MDS codes with larger lengths and dimensions, and discuss the monomial equivalence between them and the well-known Roth-Lempel codes. Some concrete examples are also given to support these results.

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Yang Li, Zhenliang Lu, San Ling, Shixin Zhu, Kwok Yan Lam. 2026-01-08. Unique Decoding of Extended Subcodes of GRS Codes Using Error-Correcting Pairs. https://arxiv.org/abs/2508.18845

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